If you are playing solo, start by realizing that you are now the full band and you need to adapt your playing like so. Think of the drummer and the bass player as "navigators" on a ship, guiding the rest of the group. So the drummer keeps time and makes fills anticipating when a period is ending and another one is beginning and the bass player plays passing notes outlining where the band is heading harmonically.
When you play solo - or when you are accompanying a solist - this is what you need to work into your play to guide both your solist and the audience.
I'm a pianist, so it hurts me to say so, but unfortunately I think both bass and drums are more important instruments than the piano. Fortunately, the piano is such a flexible instrument that - provided that you have the skills - you can actually kick out both the drummer and bass player and take over their roles, but you need to understand the responsibility you are taking when doing so.
So given that there is only piano and vocal, you would obviously play the bass in your left hand and chords (preferably root-less) in your right hand. I don't suggest playing two-hand voicings as suggested above, as this would most likely inhibit you from playing good basslines. It works well in a band setting, though. If you find it challenging to play an interesting bassline while comping, reduce the amount of comping and focus on the bass until you are comfortable enough to add more comping (if needed - you may end up finding that minimal comping sounds better after all).
Generally always play roots on the downbeat. By coincidence I stumbled upon this interview with Bootsy Collins yesterday where he explains it pretty well:
Once you have the roots nailed, start playing with adding in some 5ths and maybe alternating the root with octaves.
Generally you should still be pretty safe playing only roots and 5ths. Unfortunately things probably still sound pretty boring. This is where you start adding "passing notes" - notes anticipating the next change. Here is an example: You are playing a I->V progression from C->G, play C for 3 beats and then play F->F#->G on the last two eights in the bar landing on the downbeat of the next bar. If it is a ballad with a triplet feel (which is very common in blues) do the same with three notes on the last 3 eight note triplets in the bar (E->F->F#->G). Don't be afraid to spell out those triplets and throw in passing notes all over the place. You don't have a drummer to help you laying down time, so you have to think like a one man band taking care of both rhythm, bass and accompaniment (in that particular prioritized order).
I had a rehearsal with a band the other day, where the bass player had to cancel with short notice. We had to practice this smash hit that I hope you'll recognize in a minute, so I had to come up with a bassline to make things interesting. This is what I came up with (see if you can guess the tune):
And here is the bassline alone, slower and from a better angle:
Notice how I play exclusively roots, 5ths and chromatic steps leading towards the next root. The changes are ||: Db - Cm - Cm - F :|| if you did not already figure that out.
If you look at a jazz blues you'll find that there is an awful lot more chords than in a regular blues. But the truth is that all these chords are simply "passing chords" (just like the bass player plays passing notes) anticipating the "real" changes which is still just a simple straight blues. For example in bar 4 you usually play a II-V leading towards step IV which will be played on bar 5 as in any other blues. The bassline can incorporate these things - even if the song was written as a regular blues - most of the time you're good even though the chords don't adapt the same change - it's just another way of adding a more interesting bassline.
Finally a word of advice about time. Given that you play the piano and sing, you have a lot of freedom in terms of stretching time and playing "rubato". Note that playing rubato does not mean "play in random time". Rather it means stretch and compress time, like if you where gradually braking and winding up an old spinning record. Don't ever skip a beat. Even though the audience might not hear it, you should always be hearing all of the subdivisions in your head and simply be playing with gradually changing the tempo and maybe sometimes playing around the beat to create a free feel. Even if you play a rubato accompaniment to singer both of you should have a somewhat mutual understanding of this and maybe make some arrangements in advance and plan where who has the responsibility of time. So the singer probably has the responsibility during phrases and towards the ending of phrases, while you have a lot of freedom to play around between phrases as long as you play a clear pickup towards the next bar.